This paper represents a major step forward in the systematic and homogeneous study of Small Magellanic Cloud (SMC) star clusters and field stars carried out by applying the calcium triplet technique. We present in this work the radial velocity and metallicity of approximately 400 red giant stars in 15 SMC fields, with typical errors of about 7km/s and 0.16dex, respectively. We added to this information our previously determined metallicity values for 29 clusters and approximately 350 field stars using the identical techniques. Using this enlarged sample, we analyze the metallicity distribution and gradient in this galaxy. We also compare the chemical properties of the clusters and of their surrounding fields. We find a number of surprising results. While the clusters, taken as a whole, show no strong evidence for a metallicity gradient (MG), the field stars exhibit a clear negative gradient in the inner region of the SMC, consistent with the recent results of Dobbie et al. For distances to the center of the galaxy less than 4{deg}, field stars show a considerably smaller metallicity dispersion than that of the clusters. However, in the external SMC regions, clusters and field stars exhibit similar metallicity dispersions. Moreover, in the inner region of the SMC, clusters appear to be concentrated in two groups: one more metal-poor and another more metal-rich than field stars. Individually considered, neither cluster group presents an MG. Most surprisingly, the MG for both stellar populations (clusters and field stars) appears to reverse sign in the outer regions of the SMC. The difference between the cluster metallicity and the mean metallicity of the surrounding field stars turns out to be a strong function of the cluster metallicity. These results could be indicating different chemical evolution histories for these two SMC stellar populations. They could also indicate variations in the chemical behavior of the SMC in its internal and external regions.